The present invention relates to a method of estimating a lifetime concerning hot carrier degradation of a MOS transistor, more specifically, it relates to improvement of accuracy in lifetime estimation. The present invention relates to also a method of simulating circuit characteristic degradation caused by hot carrier degradation of a MOS transistor, more specifically, it relates to improvement of simulation accuracy.
The size of MOS transistors has been reduced considerably with a trend toward high density, high integration and miniaturization of semiconductor integrated circuit devices. With the miniaturization trend, especially due to the decreasing channel length, hot carrier degradation has been a critical problem, since the degradation will affect the reliability of a MOS transistor. Hot carrier degradation refers to a phenomenon that highly energized electrons and positive holes (hereinafter xe2x80x98hot carrierxe2x80x99) are generated by a high electric field at a drain end of a MOS transistor, which will degrade the characteristics of a gate oxide film. This hot carrier degradation includes plural degradation modes. When the degradation relates to a condition to cause a maximum substrate current, a drain current is decreased over time for any of N-type or P-type MOS transistor. As a result, the delay time of the circuit is increased over time. When the delay reaches a certain degree, a timing error occurs at a time of input/output of signals in the interior of the semiconductor integral circuit or between the circuit and outside, and this causes malfunction of an entire system in which the semiconductor integrated circuit is assembled.
Regarding the hot carrier degradation, hot carrier reliability has been evaluated by a stress acceleration experiment under a DC condition with respect to the MOS transistor. And product reliability has been provided by optimizing a production process to satisfy the hot carrier evaluation standard.
A hot carrier lifetime model used in such a hot carrier reliability evaluation is as follows. Hot carrier degradation of a MOS transistor is evaluated by, for example, xcex94Id/Id, and this is a ratio of a drain current variation xcex94Id to an initial drain current Id. Under a static hot carrier stress condition by a DC (direct current), the hot carrier degradation rate xcex94Id/Id is represented by the following formula (1).
xcex94Id/Id=Axc2x7tnxe2x80x83xe2x80x83(1)
Here, t denotes a hot carrier stress time, while characters xe2x80x98Axe2x80x99 and xe2x80x98nxe2x80x99 are regarded as coefficients depending on a transistor manufacturing process and a stress condition.
If a transistor lifetime xcfx84 is defined as a stress time required for a variation rate of drain current to reach (xcex94Id/Id)f, a formula (2) is obtained from the formula (1). For example, time t when (xcex94Id/Id)f=10% is defined as a lifetime xcfx84.
(xcex94Id/Id)f=Axc2x7xcfx84nxe2x80x83xe2x80x83(2)
In a typical stress acceleration experiment for a MOS transistor, DC stress is applied to a transistor so that the transistor lifetime reaches a variation rate (xcex94Id/Id)f defined by the formula (2) within a measureable time period, that is, from 1 second to about 100,000 seconds. Then, a drain current of the transistor is measured to calculate a transistor lifetime from xcex94Id/Id in a linear region or a saturation region.
The following stress voltage application method is used in a stress acceleration experiment during a hot carrier reliability evaluation. Every gate voltage Vg is determined at a condition where the hot carrier degradation rate is maximized with respect to each of plural drain voltages Vd used for the acceleration experiment. In other words, each of the gate voltages Vg causes a maximum substrate current Isub under a respective drain current. At this time, one gate voltage is set for each drain voltage. In this way, a transistor lifetime is calculated under a condition that the degradation rate is maximized with respect to an arbitrary drain voltage.
A method of evaluating hot carrier reliability of a MOS transistor is described in IEEE Electron Device Lett., vol. 4, pp. 111-113, April 1983 by E. Tanaka et al. According to the description, the lifetime xcfx84 of a MOS transistor is calculated by using an empirical model represented by the following formula (3).
xcfx84xe2x88x9d(Isub/W)xe2x88x92mxe2x80x83xe2x80x83(3)
In this formula, W denotes a gate width and Isub denotes a substrate current.
FIG. 6 shows a method of estimating a lifetime based on this empirical model. In FIG. 6, each black dot 21 denotes a measured value of the lifetime, and a line 22 denotes a regression line of lifetime estimation. Numeral 23 denotes a maximum substrate current value for a unit gate width in actual use, and 24 denotes an estimated lifetime in actual use. For a lifetime estimation, a logarithm of Isub/W is used to enter a horizontal axis of a graph, and a logarithm of T is used to enter the vertical axis so that the measured values 21 for a lifetime are plotted. Next, the regression line 22 is fitted to the measured values 21 by using a least squares method. The maximum substrate current 23 for a unit gate width in actual use is also measured separately. The preliminary fitted regression line 22 is used to obtain a lifetime 24 corresponding to the maximum substrate current 23 in actual use, and this is determined as an estimated lifetime in actual use. Hot carrier reliability evaluation is executed by observing whether the lifetime 24 satisfies a hot carrier evaluation standard, e.g., whether the lifetime 24 satisfies a standard of at least 10 years.
Recently however, such a conventional hot carrier evaluation standard has been difficult to satisfy in the hot carrier reliability evaluation under the DC condition. For solving this problem, a recently developed technique provides product reliability by a simulation of hot carrier degradation for a semiconductor integrated circuit (hereinafter xe2x80x9ccircuit reliability simulationxe2x80x9d). A circuit reliability simulator simulates a circuit operation subsequent to hot carrier degradation by using a hot carrier lifetime model and a SPICE parameter after degradation, and the simulation is based on calculated values of voltage and current at every terminal of every transistor which are calculated by a circuit simulator SPICE. Typical simulators are BERT developed at the University of California, Berkeley (R. H. Tu et al., xe2x80x9cBerkeley reliability tools-BERT,xe2x80x9d IEEE Trans. Compt.-Aided Des. Integrated Circuits and Syst., vol.12, no.10, pp.1524-1534, October 1993), and BTABERT (a commercial version of BERT). This circuit reliability simulation technique is used for estimating degraded or malfunctioning parts in a semiconductor integrated circuit and measures against the degradation or malfunction are taken during designing, so that reliability assurance or reliability design is possible.
An example of simulation methods concerning hot carrier degradation of a MOS transistor is described in IEEE Trans. Electron Devices, vol.35, pp.1004-1011, July 1988 by Kuo et al. A hot carrier lifetime model applied to this circuit reliability simulator is as follows. According to Kuo et al., a lifetime xcfx84 of a MOS transistor is represented by an empirical formula (4) using a hot carrier lifetime model.
xcfx84=((xcex94Id/Id)f)1/nxc2x7Hxc2x7Wxc2x7Isubxe2x88x92mxc2x7Idmxe2x88x921xe2x80x83xe2x80x83(4)
In the formula, W denotes a gate width, H denotes a coefficient depending on a condition for manufacturing a transistor, Isub denotes a substrate current, and m denotes an index relating to an impact ionization and interface level formation.
A coefficient A in a hot carrier lifetime model is represented by a formula (5) that is derived from the formulas (2) and (4).
A=((Wxc2x7H)xe2x88x921xc2x7Isubmxc2x7Id1xe2x88x92m)nxe2x80x83xe2x80x83(5)
Therefore, a formula (6) is derived from the formulas (1) and (5).
xcex94Id/Id=((Wxc2x7H)xe2x88x921xc2x7Isubmxc2x7Id1xe2x88x92mxc2x7t)nxe2x80x83xe2x80x83(6)
When Age is defined for convenience as in the following formula (7), the formula (6) can be rewritten into a formula (8).
Age=(Wxc2x7H)xe2x88x921xc2x7Isubmxc2x7Id1xe2x88x92mxc2x7txe2x80x83xe2x80x83(7)
xcex94Id/Id=(Age)nxe2x80x83xe2x80x83(8)
In the formula (7), xe2x80x98Agexe2x80x99 represents a stress quantity from a start of hot carrier stress to a time t in a hot carrier lifetime model. For a physical aspect of view, it represents a total quantity of hot carrier having energy of at least a critical energy to generate damage in a MOS transistor.
The parameters n, H and m used in the formulas (4)-(8) are regarded as hot carrier lifetime parameters. These hot carrier lifetime parameters are functions of vertical electric field strength at a drain end where the hot carrier is generated. Therefore, these parameters are represented as functions of a gate-drain voltage Vgd.
FIGS. 7A and 7B show a method of simulating characteristics after degradation by using a xcex94Id model. A simulation method using a xcex94Id model is described in IEEE Trans. Electron Devices, vol. 40, pp.2245-2254, December 1993 by Quader et al.
FIGS. 7A and 7B are equivalent circuit diagrams showing a method of simulating hot carrier degradation of a MOS transistor. In FIGS. 7A and 7B, 25 denotes a fresh MOS transistor before stress application, and 26 denotes a variable current source. FIG. 7A shows a drain current Id flowing in a fresh MOS transistor before stress application. FIG. 7B shows a drain current Idxe2x80x2 flowing in a MOS transistor after hot carrier degradation. It is shown that the drain current flowing in the transistor changes from the initial drain current Id by xcex94Id.
As shown in the following formula (9), a drain current Idxe2x80x2 after degradation is simulated by adding degradation xcex94Id of a drain current to a fresh drain current Id before stress application.
Idxe2x80x2=Id(Vd, Vg)+xcex94Id(Age, Vd, Vg)xe2x80x83xe2x80x83(9)
xcex94Id is a function of Age as stress quantity from the start of hot carrier stress to a time t, as well as a function of a drain voltage Vd and a gate voltage Vg. For calculating Age under a dynamic stress condition by AC (alternating current) in a circuit, the formula (7) is rewritten into the following formula (10) as an integral form about time for calculation.
Age=∫[(Wxc2x7H)xe2x88x921xc2x7Isubmxc2x7Id1xe2x88x92m]dtxe2x80x83xe2x80x83(10)
In this simulation, xcex94Id is represented by an equivalent circuit prepared by adding a variable current source 26 shown in FIG. 7B to a source-drain of an initial MOS transistor. At this time, a transistor parameter to calculate the initial drain current is not changed.
FIG. 8 is a flow chart to show a process to simulate hot carrier degradation of a MOS transistor according to a conventional technique. In this flow chart, a step S1 includes sub-steps S1a-S1g to extract an unknown parameter in the formulas (9) and (10) with respect to a hot carrier lifetime model by a preliminary measuring experiment.
In the sub-step S1a, a model formula Isub=g(Vg, Vd) is determined to fit to measurement data of plural substrate currents Isub in a preliminary measuring experiment, so that the substrate current Isub in the formula (10) is determined. Here, Vg denotes a gate voltage, and Vd denotes a drain voltage. An example of a method for determining a substrate current Isub is described in IEEE Electron Device Lett., vol. EDL-5, pp.505-507, December 1984 by Chan et al.
The sub-steps S1b-S1dxe2x80x2 are for extracting hot carrier lifetime parameters in a preliminary measuring experiment. In the sub-step S1b, a stress voltage is applied to a MOS transistor, and a hot carrier lifetime defined by the formula (2) is measured. For applying the stress voltage, a gate voltage Vg is set so that a gate-drain voltage Vgd=Vgxe2x88x92Vd is constant with respect to plural drain voltages Vd. In this method, typically plural numbers of Vgd are set, and also gate voltages Vg=Vd+Vgd corresponding to the plural Vgd are set with respect to every drain voltage Vd. In the following sub-step S1cxe2x80x2, coefficient n is extracted as a function of Vgd by a comparison between the empirical formula (1) and data concerning measurement points in a DC stress experiment for the sub-step S1b. Similarly in the sub-step S1dxe2x80x2, an index m and a coefficient H are extracted as functions of Vgd by a comparison between the empirical formula (4) and data concerning measuring points in a DC stress experiment for the sub-step S1b. 
Sub-steps S1e-S1g are for determining a fresh drain current Id before stress application and degradation xcex94Id of the drain current in the formula (9) for a xcex94Id model. In the sub-step S1e, transistor parameters such as carrier mobility and a flat-band voltage are extracted. Such parameters are used for determining fresh drain current Id(Vd, Vg) before stress application. BSIM (Berkeley Short-Channel IGFET Model) is used for a model to determine such a fresh drain current Id(Vd, Vg). The BSIM is described in detail in IEEE J. Solid-State Circuits, vol.SC-22, pp.558-566, August 1987 by Sheu et al. Subsequently in the sub-step S1f DC stress is applied to the transistor. In the sub-step S1g, xcex94Id model parameters are extracted before and after the DC stress application. The drain current degradation xcex94Id(Age, Vd, Vg) is determined by the xcex94Id model parameters. The xcex94Id model is described by Quader et al. in relation to NMOS. PMOS is described in JP-A-08-64814.
The transistor parameters should be extracted before DC stress application so that actual transistor characteristics before the stress application coincide with simulated transistor characteristics. The xcex94Id model parameters should be extracted before and after the DC stress application so that the actual drain current degradation xcex94Id before and after the stress application coincides with the simulated drain current degradation xcex94Id.
The step S2 includes sub-steps S2a-S2d so that a reliability simulator simulates hot carrier degradation of a transistor in accordance with parameters extracted in the step S1 and also with the formulas (9) and (10).
In the sub-step S2a, a drain current is simulated by transistor parameters before stress application, where the parameters have been extracted in the prior sub-step S1e. In the sub-step S2b, a substrate current is simulated on the basis of a substrate current model formula Isub=g(Vg, Vd) determined by the S1a. In the sub-step S2cxe2x80x2, Age, which represents degradation of each transistor based on the formula (10), is calculated by time-integrating functions of a drain current Id and a substrate current Isub in a circuit. At this time, the drain current Id simulated in the sub-step S2a, the substrate current Isub simulated in the sub-step S2b, and hot carrier lifetime parameters H and m calculated in the sub-step S1dxe2x80x2, are used. In the sub-step S2d, hot carrier degradation of the transistor is simulated by using the formula (9) on the basis of the Age.
The following is a detailed explanation of a method of extracting hot carrier lifetime parameters H and m for a hot carrier lifetime model. FIG. 9 is an explanatory view of a method of extracting hot carrier lifetime parameters H and m. FIG. 9 relates to a plot for calculating hot carrier lifetime parameters H and m included in the empirical formula (4) using a hot carrier lifetime model. In FIG. 9, the vertical axis is a logarithmic scale of a value xcfx84xc2x7Id/W calculated from a lifetime xcfx84 in a DC stress experiment, a drain current Id during a stress and a gate width W of a MOS transistor. The horizontal axis is a logarithmic scale of a ratio Isub/Id when Isub is a substrate current during a stress and Id is a drain current. Numeral 27 denotes data concerning a plurality of measurement points in a DC stress experiment, and 28 denotes a line fitted with respect to data concerning the measurement points. The MOS transistor lifetime xcfx84 is measured under plural gate-drain voltage Vgd conditions, e.g., under three conditions of Vgd=0.0, xe2x88x921.0, and xe2x88x922.0V, so that data 27 for plural measurement points are obtained. In this manner, a line 28, fitted by a least squares method with respect to data 27 concerning the measurement points, is obtained. Hot carrier parameters H and m are obtained respectively from an intercept and a gradient of the line 28. By executing this method for plural Vgd, the hot carrier lifetime parameters H and m in a hot carrier lifetime model can be calculated as functions of Vgd.
The above description is about a conventional method of estimating lifetime of hot carrier degradation of a MOS transistor, and a conventional method of simulating degradation in circuit characteristics caused by the hot carrier degradation. However, these methods can cause the following problems.
First, a lifetime under a condition for causing a maximum hot carrier degradation rate can be estimated to be longer than an actual lifetime in a method of estimating a lifetime of the hot carrier degradation of a MOS transistor. In such a method, it is hypothesized that a gate voltage causing maximum hot carrier degradation coincides with a gate voltage causing a maximum substrate current. In an empirical model (3), degradation is maximized as well at a gate voltage causing a maximum substrate current. Actually however, some processes for manufacturing transistors have maximum degradation at a gate voltage lower than a gate voltage causing a maximum substrate current. Therefore, lifetime in actual use can be shortened depending on a condition to use the MOS transistor as opposed to a lifetime corresponding to the maximum substrate current estimated according to this model. This results in a problem that the quality assurance of the product cannot be provided properly.
Concerning a method of simulating circuit characteristic degradation caused by the hot carrier degradation, numbers of transistors are required to calculate hot carrier lifetime parameters, and this method will take a long period of time. A DC stress experiment should be executed by setting plural gate voltages Vg in order to keep a gate-drain voltage Vgd=Vgxe2x88x92Vd to be constant with respect to plural drain voltages Vd, so that the index m and the coefficient H in the empirical formula (4) representing a conventional hot carrier lifetime model are extracted as functions of Vgd. In this case, total numbers of stress voltage conditions are increased considerably. For example, about three conditions for a drain voltage, and about five conditions for a gate-drain voltage are required, which results in about 15 stress voltage conditions. Moreover, since the results of the DC stress experiment vary, plural (e.g., about three) transistors are required for every voltage condition in order to obtain sufficiently accurate results. As a result, about 45 transistors will be required. In addition to that, these transistors should be produced under an identical process condition. Since prototypes are produced repeatedly while altering the process condition during a process development, it is difficult to prepare such large numbers of transistors under an identical process condition. Moreover, measurement of 45 transistors under 15 voltage conditions will take about 10 days with ordinary equipment and staff. As a result, it is difficult to provide prompt feedback to the product designing. Since the product is designed in parallel with the process development, substantially an actual process development has not carried out a step of extracting a hot carrier lifetime parameter to simulate reliability in order to realize product reliability at the time of designing. As a result, a standard of process reliability evaluation with an excessive reliability margin is applied for quality assurance. It is difficult to meet such a reliability evaluation standard while providing high transistor performance.
The cause of requiring numbers of transistors and long period of time for an experiment to calculate a hot carrier lifetime parameter is that a conventional hot carrier lifetime model represented with the formula (4) is insufficient for the purpose. Such a conventional hot carrier lifetime model is based on a hypothesis that hot carrier degradation is caused by a hot carrier of either a positive hole or an electron having energy of at least a critical energy required for generating damage to a MOS transistor, and that a hot carrier lifetime is inversely proportional to a yield of the one hot carrier. In this model, no functional forms are provided with respect to dependence of a gate-drain voltage Vgd of the hot carrier lifetime parameters H and m. Therefore, a DC stress experiment should be performed by setting plural gate voltages Vg with respect to plural drain voltages Vd in order to measure the Vgd dependence of the parameters H and m.
In order to solve the above-identified problems, the present invention provides a method of estimating lifetime of hot carrier degradation of a MOS transistor, so that the lifetime is estimated with accuracy under a condition for causing a maximum hot carrier degradation rate, and thus, the present invention provides proper quality assurance for a product.
The present invention enables calculating a hot carrier lifetime parameter in a short time by using a small number of transistors in a method of simulating circuit characteristic degradation caused by hot carrier degradation in order to provide product reliability at a time of designing, and to provide high transistor performance.
For the above-mentioned purposes, a method of estimating a lifetime of a semiconductor device according to the present invention is characterized in that a hot carrier lifetime is estimated depending on a hot carrier lifetime model:
xcfx84xe2x88x9dIsubxe2x88x92mxc2x7Idmxe2x88x922
where xcfx84 is a lifetime, Isub is a substrate current, Id is a drain current, and m is a fitting parameter.
The above-described hot carrier lifetime model according to the present invention reflects the fact that hot carrier degradation occurs due to two kinds of hot carriers of an electron and a positive hole having an energy of at least a critical energy required for generating damage to a MOS transistor, and that the hot carrier lifetime is inversely proportional to a product of a yield of these two hot carriers. According to this model, a model formula representing a hot carrier lifetime takes not either the conventional formulas (3) or (4) but the following formula (11).
xcfx84=((xcex94Id/Id)f)1/nxc2x7Hxc2x7W2xc2x7Isubxe2x88x92mxc2x7Idmxe2x88x922xe2x80x83xe2x80x83(11)
The model formula (11) is distinguished from the conventional model formula (4) in that the formula (11) adopts a functional form of (exponent of Id)=xe2x88x92(exponent of Isub)xe2x88x922, while the conventional formula adopts (exponent of Id)=xe2x88x92(exponent of Isub)xe2x88x921. The difference is caused by the fact that a hot carrier lifetime is inversely proportional to a yield of one kind of hot carrier in a conventional technique, while the same lifetime is inversely proportional to a yield of two kinds of hot carriers in the present invention.
It was confirmed by a comparison with measured values of N-type and P-type MOS transistors that a model of the present invention corresponds well to a measured value, and Vgd dependence of the hot carrier lifetime parameters H and m is decreased considerably. The reason is considered to be that the model of the present invention reflects accurately a mechanism of hot carrier degradation of a MOS transistor. In a conventional model, Vgd dependence of the hot carrier lifetime parameters H and m is increased. The reason is considered to be that such a model failed to reflect accurately a mechanism of hot carrier degradation of a MOS transistor.
In the above-mentioned method of estimating a lifetime of a semiconductor device, a minimum value of a hot carrier lifetime in actual use can be estimated from a substrate current and a drain current under a use condition. More specifically, the method of estimating lifetime of hot carrier degradation according to the present invention is executed in the following manner. First, the hot carrier lifetime parameters H and m in the formula (11) are calculated as constants based on measured values in a stress acceleration experiment. These parameters and formula (11) are used for obtaining a condition for causing maximum hot carrier degradation in actual use based on measured values of a substrate current Isub and a drain current Id in actual use, and a hot carrier lifetime at that time also is obtained. Thus obtained lifetime is applied as an estimated lifetime in actual use.
A method of simulating the reliability of a semiconductor device according to the present invention is characterized in that a parameter Age representing cumulative stress quantity relating to a MOS transistor is calculated on the basis of the following formula for simulating hot carrier degradation of the MOS transistor:
Agexe2x88x9d∫[Isubmxc2x7Id2xe2x88x92m]dt
where t is time, Isub is a substrate current, Id is a drain current, and m is a fitting parameter.
For example, calculation is performed by using the following formula (12) as a functional form for time in place of the formula (10) in a method of simulating circuit characteristic degradation caused by hot carrier degradation of a MOS transistor.
Age=∫[(W2xc2x7H)xe2x88x921xc2x7Isubmxc2x7Id2xe2x88x92m]dtxe2x80x83xe2x80x83(12)
In extraction of hot carrier lifetime parameters, hot carrier lifetime parameters H and m are calculated as constants based on measured values in a stress acceleration experiment. Here, the model formula (12) of the present invention is distinguished from the conventional model formula (10) in that the model formula (12) adopts a functional form of (exponent of Id exponent)=2xe2x88x92(exponent of Isub), while the conventional formula adopts (exponent of Id)=1xe2x88x92(exponent of Isub). The difference reflects the fact that a hot carrier lifetime is inversely proportional to a yield of one kind of hot carrier in a conventional technique, while the same lifetime is inversely proportional to a yield of two kinds of hot carriers in the present invention. The hot carrier lifetime parameters H and m are identical to those in a model formula (11) for a hot carrier lifetime. Vgd dependence is considerably lowered, and thus, sufficient simulation accuracy can be obtained even if the parameters are applied as constants.
This model formula (11) is obtained in the following manner. This model formula (11) is derived from a lucky electron model (IEEE Electron Devices, vol. ED-32, pp. 375-385, February 1985) and a hypothesis composing a physical logic of the present invention, i.e., xe2x80x9chot carrier degradation occurs due to two kinds of hot carriers of an electron and a positive hole having energy of at least a critical energy required for generating damage to a MOS transistor, and the hot carrier lifetime is inversely proportional to a product of a yield of these two kinds of hot carriersxe2x80x9d.
For NMOS, Ie, Ih, and Isub are represented respectively by formulas (13), (14), and (15), in which Ie is an electron yield having energy of at least a critical energy required for generating damage to a MOS transistor, Ih is a yield of a positive hole having energy of at least a critical energy, and Isub is a substrate current.
Ie=Id exp(xe2x88x92"PHgr"e/qxcexeEm)xe2x80x83xe2x80x83(13) 
Ih=Isub exp(xe2x88x92"PHgr"h/qxcexhEm)xe2x80x83xe2x80x83(14) 
Isub=Id exp(xe2x88x92"PHgr"ei/qxcexeEm)xe2x80x83xe2x80x83(15)
Here, Id is a drain current, Isub is a substrate current, "PHgr"e is a critical energy of an electron required for generating damage, "PHgr"h is a critical energy of a positive hole required for generating damage, "PHgr"ei is a critical energy of an electron required for impact ionization, xcexe is an electron mean free path, xcexh is a mean free path of a positive hole, and Em is a maximum channel electric field strength. When a lifetime xcfx84 is hypothesized as being inversely proportional to a product of Ie/W and Ih/W, and qEm is deleted by applying the formulas (13), (14) and (15), the following formula (16) is obtained. In this case, Ie/W is a yield per unit gate width of an electron having energy of at least a critical energy, and Ih/W is a yield of a positive hole having energy of at least a critical energy.                                                         τ              ∝                              xe2x80x83                            ⁢                                                W                  2                                /                                  (                                                            I                      e                                        ·                                          I                      h                                                        )                                                                                                        =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                      (                                                                  I                        d                                            ·                                              I                        sub                                                              )                                                  ⁢                                  exp                  ⁡                                      [                                                                  (                                                                                                            Φ                              h                                                        /                                                          λ                              h                                                                                +                                                                                    Φ                              e                                                        /                                                          λ                              e                                                                                                      )                                            /                                              qE                        m                                                              ]                                                                                                                          =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                      (                                                                  I                        d                                            ·                                              I                        sub                                                              )                                                  ⁢                                                      (                                                                  I                        sub                                            /                                              I                        d                                                              )                                    ⋀                  -                                                                                                                        xe2x80x83                            ⁢                              [                                                      (                                                                                            Φ                          h                                                /                                                  λ                          h                                                                    +                                                                        Φ                          e                                                /                                                  λ                          e                                                                                      )                                    /                                      (                                                                  Φ                        ei                                            /                                              λ                        e                                                              )                                                  ]                                                                        (        16        )            
As a result of replacement as shown in the following formula (17),
m=("PHgr"h/xcexh+"PHgr"e/xcexe)/("PHgr"ei/xcexe)+1xe2x80x83xe2x80x83(17)
a formula (18) is obtained from the formula (16).
xcfx84xe2x88x9d(W/Id)2(sub/Id)xe2x88x92mxe2x80x83xe2x80x83(18)
When the parameter H is determined considering the formula (2), the model formula (11) representing a hot carrier lifetime of the present invention is obtained.
The way to derive the conventional formula (4) is described below in order to explain why the exponents of Id are different between the conventional model formula (4) and the model formula (11). The conventional model formula (4) depends on a hypothesis that hot carrier degradation occurs only due to an electron having energy of at least a critical energy required for generating damage to a MOS transistor, and that a hot carrier lifetime is inversely proportional to a yield of this hot carrier. When this hypothesis is adopted, a formula (19) corresponding to the formula (16) is obtained as follows.                                                         τ              ∝                              xe2x80x83                            ⁢                                                W                  2                                /                                  I                  e                                                                                                        =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                      I                    d                                                  ⁢                                  exp                  ⁡                                      (                                                                                            Φ                          e                                                /                        q                                            ⁢                                              xe2x80x83                                            ⁢                                              λ                        h                                            ⁢                                              E                        m                                                              )                                                                                                                          =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                                            I                      d                                        ⁡                                          (                                                                        I                          sub                                                /                                                  I                          d                                                                    )                                                                      ⋀                                  -                                      xe2x80x83                                    ⁢                                      [                                          (                                                                        Φ                          e                                                /                                                  Φ                          ei                                                                    )                                        ]                                                                                                          (        19        )            
As a result of replacement as shown in the following formula (20),
m="PHgr"e/"PHgr"eixe2x80x83xe2x80x83(20)
a formula (21) is obtained from the formula (19).
xcfx84xe2x88x9dW/Id(Isub/Id)xe2x88x92mxe2x80x83xe2x80x83(21)
As mentioned above, there is a difference in the exponents of Id between the model formula (18) of the present invention and the conventional model formula (21). A difference in formation of model formulas reflects the difference in the hypothesis about what the lifetime is inversely proportional to.
For PMOS, Ie, Ih, and Isub are represented respectively by formulas (22), (23), and (24), in which Ie is an electron yield having energy of at least a critical energy required for generating damage to a MOS transistor, Ih is a yield of a positive hole having energy of at least a critical energy, and Isub is a substrate current.
Ie=Isub exp(xe2x88x92"PHgr"e/qxcexeEm)xe2x80x83xe2x80x83(22) 
Ih=Id exp(xe2x88x92"PHgr"h/qxcexhEm)xe2x80x83xe2x80x83(23) 
Isub=Id exp(xe2x88x92"PHgr"hi/qxcexhEm)xe2x80x83xe2x80x83(24)
Here, Id is a drain current, Isub is a substrate current, "PHgr"e is a critical energy of an electron required for generating damage, "PHgr"h is a critical energy of a positive hole required for generating damage, "PHgr"hi is a critical energy of a positive hole required for impact ionization, xcexe is an electron mean free path, xcexh is a mean free path of a positive hole, and Em is a maximum channel electric field strength. When a lifetime xcfx84 is hypothesized as being inversely proportional to a product of Ie/W and Ih/W, and qEm is deleted by applying the formulas (22), (23) and (24), the following formula (25) is obtained. In this case, Ie/W is a yield per unit gate width of an electron having energy of at least a critical energy, and Ih/W is a yield of a positive hole having energy of at least a critical energy.                                                         τ              ∝                              xe2x80x83                            ⁢                                                W                  2                                /                                  (                                                            I                      e                                        ·                                          I                      h                                                        )                                                                                                        =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                      (                                                                  I                        d                                            ·                                              I                        sub                                                              )                                                  ⁢                                  exp                  ⁡                                      [                                                                  (                                                                                                            Φ                              h                                                        /                                                          λ                              h                                                                                +                                                                                    Φ                              e                                                        /                                                          λ                              e                                                                                                      )                                            /                                              qE                        m                                                              ]                                                                                                                          =                              xe2x80x83                            ⁢                                                                    W                    2                                    /                                      (                                                                  I                        d                                            ·                                              I                        sub                                                              )                                                  ⁢                                                      (                                                                  I                        sub                                            /                                              I                        d                                                              )                                    ⋀                  -                                                                                                                        xe2x80x83                            ⁢                              [                                                      (                                                                                            Φ                          h                                                /                                                  λ                          h                                                                    +                                                                        Φ                          e                                                /                                                  λ                          e                                                                                      )                                    /                                      (                                                                  Φ                        hi                                            /                                              λ                        h                                                              )                                                  ]                                                                        (        25        )            
When the above formula is rewritten as follows,
m=("PHgr"h/xcexh+"PHgr"e/xcexe)/("PHgr"hi/xcexh)+1xe2x80x83xe2x80x83(26)
a formula (18) similar to that of NMOS is obtained from the formula (25). When the parameter H is determined considering the formula (2) as in the case of NMOS, the model formula (11) representing a hot carrier lifetime of the present invention is obtained.
The formula (12) for calculating Age, which is used in a method of simulating circuit characteristic degradation caused by hot carrier degradation of a MOS transistor of the present invention, is derived from a comparison with the formula (8) after parameters A and xcfx84 are deleted from the formulas (11) together with the formulas (1) and (2).
In the model formula (11) of the present invention, the gate-drain voltage Vgd dependence of the hot carrier lifetime parameter is extremely small for N-type and P-type MOS transistors. Therefore, even if it is used as a constant, a lifetime can be estimated accurately in a wide range of voltage. As a result, in a method of estimating lifetime of hot carrier degradation of a MOS transistor of the present invention, a voltage condition for causing a maximum degradation rate and the lifetime, i.e., a lifetime in actual use, can .be estimated accurately from a measured value under a small number of stress voltage conditions and a drain current Id and a substrate current Isub in actual use. This serves to provide proper quality assurance of the product.
Since the hot carrier lifetime parameters H and m are constants in a method of simulating circuit characteristic degradation caused by hot carrier degradation of a MOS transistor according to the present invention, the hot carrier lifetime parameters H and m can be obtained from measured values taken under a small numbers of stress voltage conditions. For example, typically three conditions are used for a drain voltage, and one condition of a gate voltage Vg causing a maximum substrate current Isub with respect to each drain voltage. In such a case, only about three conditions will be required for the stress voltage conditions. Since results of a DC stress experiment vary, plural numbers of (e.g., about three) transistors should be used for each voltage condition in order to obtain sufficient accuracy. Even so, only about nine transistors are required. In other words, the number of transistors can be decreased considerably from the numbers of 45 in a conventional technique. When about 9 transistors of the same process conditions are prepared for measurement under three voltage conditions, it will take only 2 days with ordinary equipment and staff. Therefore, a prompt feedback to the product designing can be provided. This enables extracting hot carrier lifetime parameters during a process development and to perform reliability simulation in order to provide product reliability at a time of designing. Therefore, it is not necessary to apply a process reliability evaluation standard having an excessive reliability margin for the purpose of quality assurance. In this way, the present invention can satisfy a reliability evaluation standard and also realize high transistor performance.
In the above-mentioned method of simulating reliability of a semiconductor device, the fitting parameter m can be a function of a gate-drain voltage, so that a simulation with higher accuracy can be performed.
Alternatively the fitting parameter m can be used as a function of the substrate voltage, so that a simulation with a high accuracy is obtainable even when the substrate voltage dependence of the hot carrier degradation is not negligible.
Furthermore in the above-mentioned method of simulating reliability of the semiconductor device, measurement data is plotted by using a logarithmic scale of a value xcfx84xc2x7(Id/W)2 calculated from a lifetime xcfx84 in a DC stress experiment, a drain current Id during a stress, and a gate width W of the MOS transistor to enter a vertical axis and by using a logarithmic scale of a ratio Isub/Id when Isub is a substrate current during stress and Id is a drain current, so that a fitting parameter m is obtained from a gradient of a line fitted with respect to the plot.